The shifted harmonic oscillator and the hypoelliptic Laplacian on the circle

TL;DR

This paper analyzes the hypoelliptic Laplacian on the circle, describing its semigroup extension, spectral properties, and providing new integral formulas and asymptotics for Laguerre polynomials.

Contribution

It introduces a novel integral formula for spectral projections and characterizes the domain of the semigroup extension using harmonic oscillator decompositions.

Findings

Boundedness in a half-plane linked to eigenfunction expansion convergence

New integral formula for spectral projections

Asymptotic analysis of Laguerre polynomials in large-parameter regimes

Abstract

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half-plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large-parameter regime.